Professor Timothy Trudgian, UNSW Canberra
Dr Valeriia Starichkova, UNSW Canberra
Dr Michaela Cully-Hugill, UNSW Canberra
We will cover several topics in Number Theory. Our main goal will be to prove the Dirichlet theorem for prime numbers in arithmetic progressions. Within the course, we will also cover other related topics. Participants will develop problem-solving skills through various exercises and motivating discussions during the workshops. Participants will get acquainted with some of the basic techniques used nowadays in analytic number theory.
- Elementary proofs about primes in arithmetic progressions.
- Primes congruent to 1 modulo k.
- Elementary attempts to prove the Dirichlet theorem via methods by Chebyshev and Erdős.
- Obstructions of elementary methods.
- Introduction to complex analysis and analytic techniques covering the Riemann zeta-function and its extension to the complex plane using the functional equation.
- An analytic proof of the infinitude of prime numbers.
- Introduction to the proof of the Dirichlet theorem: the Dirichlet characters and the properties of the Dirichlet L-functions.
- Finalising the proof of the Dirichlet theorem.
- More questions around primes represented by polynomials and primes in arithmetic progressions like the Green-Tao theorem.
- Arithmetic progressions; acquaintance with modular arithmetic; complex numbers; one-variable calculus.
- A basic understanding of complex analysis would be helpful but not required.
- Problem-solving (60%): Students will solve exercises and present them to lecturers or workshop demonstrators during the workshops. May include some quizzes conducted in workshops.
- Final exam at the end of week 4 (40%)
(subject to change)
Participation in all lectures and tutorials is expected.
For those completing the subject for their own knowledge/interest, evidence of at least 80% attendance at lectures and tutorials is required to receive a certificate of attendance.
No preliminary reading is required, but the books below will be helpful for those who would like to get acquainted with the techniques earlier.
- “An Introduction to the Theory of Numbers” by G.H. Hardy and E.M. Wright; or
- “Multiplicative Number Theory” by H. Davenport
Good resources for practice are:
- “Problems in Analytic Number Theory” by M.R. Murty; or
- “Steps into Analytic Number Theory” by P. Pollack and A. Singha Roy.
Not sure if you should sign up for this course?
Take this QUIZ to self-evaluate and get a measure of the key foundational knowledge required.